Given a substochastic kernel P from a measurable space (E,β) into itself one considers for a pair (μ,ν) of finite measures on β the following sequences:
μ0=(μ-ν)+,ν0=(μ-ν)-;μn+1=(μnp-νn)+,νn+1=(μnp-νn)-,n=0,1,2,...
This paper deals with conditions for ↓limnνn=0, or limn∥μn∥=0 to hold. As an application a characterization of those measures ν is given which may occur in a P-Markov chain (Xn)n∈N with state space E, having μ as its initial law, as distribution of Xτ where τ is a suitable stopping time.